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Suppose you have a pool table that measures $165\times297$. If you shoot a ball from the lower left corner at a $45^\circ$ angle, assume it will continue moving until it lands in a corner pocket and that each time it hits the wall, it will bounce off at a $45^\circ$ angle. Also, assume there are only corner pockets.

How many bounces will it take before it goes into a pocket?

Please show the work and patterns you found, and also please include a general rule.

I have tried multiple times to find where the ball bounces too but I keep getting $4$.

My equation is $\frac{l}{GCF} + \frac {h}{GCF} - 2$ but I can't seem to figure out the answer to the problem.

2 Answers 2

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When the ball rebounds on an edge, reflect the entire board on that edge. Then the path of the ball is the straight line extension to the reflected board. Thus when the ball lands in a corner of the board, the number of horizontal reflections $m$ and the number of vertical reflections $n$ must satisfy $297 m = 165 n$. Thus $m=5, n=9$. The ball lands in a corner after $14$ rebounds.

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    please could you do the problem again with 297 instead of 197, my apologies – 2017-02-16
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    More generally, if the table size is $a \times b$ and the ball is projected at an angle $\theta$ to the horizontal, then the ball lands in a corner if and only if there are integers $m,n$ such that $\tan \theta = \frac{ma}{nb}$. In the special case where $a,b$ are integers and $\tan \theta$ is not rational, the ball goes on for ever and the path of the ball covers every point on the board eventually. – 2017-02-16
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Well with the 45 degrees we have an isosceles triangle. Thus the ball will hit 165 up the right side (197-165=32 from the upper right pocket). Then it hits the top side 165-32=133 from the upper left pocket. Then it hits the left side 197-133=64 from the bottom left pocket. Then 165-64=1 from the bottom right.. this goes on and on until you land in a pocket... having no luck coming up with any clever way to write a general formula. But I think this will eventually get you there.

Just read the other guys explanation its betters

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    please could you do the problem again with 297 instead of 197, my apologies – 2017-02-16
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    Well from my answer you would just put 297 instead of 197 where I wrote "(197-165=32 from the upper right pocket)" You would get 297-165=132 then 165-132=133 then 297-133= 164 and so on.. But using Muralidharn's approach which is far more elegant you would have 297m= 165n thus m=165 and n =297 then 165+297 = 432 rebounds – 2017-02-16